Is it possible to unwrap genus-0 mesh so each vert has one and only unique uv?
No.

Unwrap. Meaning peeling the surface off of a solid and laying it out flat on the ground. 

If your requirement is that there are no duplicated texture coordinates at all, it would mean that you are trying to peel an orange without cutting its skin. Even if you manage to "beam off" the skin in one piece you end up with a hollow shell that you cannot lay out flat on the ground without the front and back sharing the same area. 

And the moment you cut it you get seams and therefore more uvs than vertices. if you merge them again later you once more end up in the situation described above.

This is true for all solids, not just genus-0 ones like a sphere.